Notes on a mathematical interpretation of the opening of Hegel’s Science of Logic

Football manager Jose Mourinho expresses his disdain for those that haven’t read Hegel.

A turn to Hegel is often a palliative when troublesome reality upsets the relationship between our theory and practice, whether that be losing football matches or losing members from your preferred revolutionary party. A good dose of Hegelian dialectics might convince a dwindling number of the faithful that, despite all evidence to the contrary, the organisation is objectively in rude health due to its access to profound insights that only a select few can understand.

So I feel to have to begin with an apology. Although I’m neither a football manager, nor responsible for leading a vanguard, I am very interested in what Hegel might have to tell us, even after all these years.

This is a long post, very abstract, and probably not to everyone’s taste. But the conclusion yields a surprising and unexpected materialist twist.

The difficulty with Hegel

Hegel’s Science of Logic, written in the early 1800s, is a difficult book, to say the least. One difficulty is that Hegel’s project seems fantastical: he aims to discover the fundamental structure of everything from pure reflection alone. Materialists and empiricists will rightly hesitate. Another difficulty is Hegel’s methodology, which is shocking: doubting even Rene Descartes’ ‘I’ he claims to start his enquiry with zero assumptions – and yet derives multiple propositions. How is it possible to reason with no axioms or inference rules? Another difficulty is Hegel’s language: difficult ideas often necessitate technical terms and complex locutions, but Hegel twists the reader up, down, left and right and the effect is dizzying. So it’s a difficult book, and one which most readers, over the centuries, have either never picked up, or quickly put down.

But I’m intrigued by Hegel’s Logic, and I think there is something in it. For instance, if you’re a materialist and believe we’re constructed from the same ‘stuff’ as everything else in the universe, then our cognition must share fundamental properties in common with everything else (let’s call those properties the ‘metaphysical bedrock’). That bedrock might be very thin indeed, but it has to be there (if you’re a materialist). In consequence, simply reflecting on the immediate properties of our own thought might give us access to these fundamental properties, and therefore be a source of knowledge. So I don’t dismiss Hegel’s project as entirely fantastical. Furthermore, dismissing philosophical reflection on the grounds that it isn’t horny-handed labour is a very vulgar kind of materialism indeed, which should have no place in the Marxist tradition.

Of course, we can’t deduce anything without assumptions and inference rules. But what if Hegel isn’t trying to do that? Perhaps by shedding ourselves of all assumptions and merely observing and understanding ‘what remains’ yields immediate access to a complex bedrock with discernible structure. We may need to observe and understand at some length, and consider the bedrock from many different angles, but we wouldn’t be performing any kind of logical deduction; instead we’d be pursuing a kind of empirical investigation in the hope that the bedrock might eventually disclose its complex of properties to us. Granted, this seems unlikely, but not prima facie impossible.

And Hegel’s language might be difficult because the task of observing and describing the metaphysical bedrock is so uncommon that our current theories and language simply don’t have the concepts or words. So the task forces Hegel to borrow existing words and apply them to this alien domain, with all the unavoidable semantic slippage and potential confusion. If so, then it should be possible to pursue Hegel’s project, and adopt his methodology, yet use alternative language frameworks. At best, this might yield new insights into the bedrock; at worst we’d hope to better understand what on earth Hegel was trying to say, and whether any sense can be made of it. So let’s try it!

In the beginning there was pure being …

Hegel writes:

Thus the beginning must be an absolute, or what is synonymous here, an abstract beginning; and so it may not suppose anything, must not be mediated by anything nor have a ground; rather it is to be itself the ground of the entire science. Consequently, it must be purely and simply an immediacy, or rather merely immediacy itself. Just as it cannot possess any determination relatively to anything else, so too it cannot contain within itself any determination, any content; for any such would be a distinguishing and an inter-relationship of distinct moments, and consequently a mediation. The beginning therefore is pure being.

So we start at the metaphysical bedrock (Hegel calls it ‘the ground’) that is entirely abstract. Here we must rid ourselves of all presuppositions, and consider what everything must have in common. And that must be existence itself, ‘pure being’.

We don’t have the Cartesian ‘I think, therefore I am’ here because we cannot assume there is an ‘I’ or even any ‘thinking’ as commonly understood. No, just pure being. And, since we have no knowledge whatsoever, this pure being ‘cannot possess any determination’ and lacks any content whatsoever. Hegel explicitly mentions the ego-less state of one who mediates and looks ‘only at the tip of his nose’ and says ‘inwardly only Om, Om, Om, or else nothing at all’. So Hegel’s starting point is genuinely trippy — it’s a psychedelic starting point.

Hegel suggests, in the context of discussing that pure being is a kind of “pure knowing”, that:

If pure being is taken as the content of pure knowing, then the latter must stand back from its content, allowing it to have free play and not determining it further. Or again, if pure being is to be considered as the unity into which knowing has collapsed at the extreme point of its union with the object, then knowing itself has vanished in that unity, leaving behind no difference from the unity and hence nothing by which the latter could be determined. Nor is there anything else present, any content which could be used to make the beginning more determinate.

So Hegel declares that pure being is self-referential where our thought contemplates its own thought, where our ‘pure knowing’ has achieved a ‘union with the object’ and where that object is itself the content of ‘pure knowing’. So we have thought contemplating its own thought – but without any presuppositions, and therefore we also abstract from ourselves, the knower, the Cartesian ego, and ‘stand back’ from this content.

Pictures help. So we can draw Hegel’s self-referential starting point in terms of pure being interacting with itself:

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Pure being: thought contemplating itself

The arrow in this diagram indicates the self-referential nature of Hegel’s beginning. To be poetic for a moment, the outward port is ‘pure being’ and the inward port is ‘pure knowing’, where being contemplates itself. Pure being has no content, and there’s nothing else to contrast it with, so what ‘flows’ along the arrow (if anything at all) must also be purely being.

… and also pure nothing

As we might expect, this doesn’t seem to be an auspicious start, since it’s not obvious anything follows at all from this psychedelic insight. But then Hegel observes:

Being, pure being, without any further determination. In its indeterminate immediacy it is equal only to itself. It is also not unequal relatively to an other; it has no diversity within itself nor any with a reference outwards. It would not be held fast in its purity if it contained any determination or content which could be distinguished in it or by which it could be distinguished from an other. It is pure indeterminateness and emptiness. There is nothing to be intuited in it, if one can speak here of intuiting; or, it is only this pure intuiting itself. Just as little is anything to be thought in it, or it is equally only this empty thinking. Being, the indeterminate immediate, is in fact nothing, and neither more nor less than nothing.

So pure being is so pure, so lacking in any content, it is in fact nothing at all! At first glance it was existence, but this kind of existence is nothing – it is empty, a complete void, nullity.

And the fact we began with pure being wasn’t necessary. We could equally have started by considering what is not in common with anything which, on reflection, must of course be nothing (since anything other than nothing would at least have something in common with something).

So we can equally draw:

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Pure nothing: thought contemplating nothing

Nothing has changed in this diagram except our label, which is now ‘nothing’. What we called pure being we now also call pure nothing.

There is a difference, however. Being has the connotation of something that exists, but pure being has no content, and so the content that exists is pure nothing. And pure nothing has the connotation that nothing exists.

This seems like a step backwards … we thought we were observing pure being – the metaphysical bedrock that remains once we abstract from all possible content – but we find that, as soon as we do that, we’re left with nothing at all.

The metaphysical bedrock is a paradox

But Hegel immediately observes that:

Nothing, pure nothing: it is simply equality with itself, complete emptiness, absence of all determination and content β€” undifferentiatedness in itself. In so far as intuiting or thinking can be mentioned here, it counts as a distinction whether something or nothing is intuited or thought. To intuit or think nothing has, therefore, a meaning; both are distinguished and thus nothing is (exists) in our intuiting or thinking; or rather it is empty intuition and thought itself, and the same empty intuition or thought as pure being. Nothing is, therefore, the same determination, or rather absence of determination, and thus altogether the same as, pure being

In other words, although pure nothing is nothing at all, it does exist, and therefore is being – in fact, it exists as pure being since it lacks any content at all. So the concept of pure nothing seems to immediately turn back into pure being. We are forced to swap back-and-forth between two viewpoints, which seem to be essentially the same, and differ only in name or label.

Now Hegel throws this remarkable paragraph at us:

Pure Being and pure nothing are, therefore, the same. What is the truth is neither being nor nothing, but that being β€” does not pass over but has passed over β€” into nothing, and nothing into being. But it is equally true that they are not undistinguished from each other, that, on the contrary, they are not the same, that they are absolutely distinct, and yet that they are unseparated and inseparable and that each immediately vanishes in its opposite. Their truth is therefore, this movement of the immediate vanishing of the one into the other: becoming, a movement in which both are distinguished, but by a difference which has equally immediately resolved itself.

Hegel, in this paragraph, appears to simultaneously pull a rabbit out of a hat (‘becoming’) while trashing the conventions of formal logic (pure being and pure nothing are ‘the same’ and ‘they are not the same’) while also introducing change (‘vanishing’, ‘movement’, ‘becoming’).

We will unpack Hegel’s paragraph, and try to make sense of it, but it will require a surprising amount of work.

Hegel, in his defence, would claim that it’s the phenomena that is doing the trashing. Since we’re at a philosophical stage of analysis, devoid of all assumptions, we can’t impose the axioms and inference rules of formal logic. We must merely observe the bedrock, and see what it’s like, and reserve judgement. Hegel says that it would be ‘dogmatic’ and unjustified if we immediately applied our current standards of logic (e.g., law of non-contradiction) at this stage of analysis. Yes, we’d like to eventually derive the necessity of such standards, but simply assuming their truth and applying them to the bedrock would be a mistake: much like the earnest scientists in sci-fi movies who setup their sophisticated measuring equipment to analyse a newly found alien artefact, only to eventually realise, in the final twist, that all their theories and presuppositions were false and in vain. Hegel wants to avoid this error (if he can).

So, according to Hegel, we have the following bedrock phenomena: pure being, which for shorthand we’ll call ‘x’, and pure nothing, which we’ll call ‘y’. Now, x and y are the same. Also, x and y are not the same. And, furthermore, x vanishes into its opposite y, and y vanishes into its opposite x. This seems tricky, perhaps even absurd.

But definitely fun to think about …

The logical structure of Hegel’s beginning is similar to self-referential paradoxes in natural language, such as Epimenides paradox. Consider the statement: “This sentence is false”. If the sentence is false then the sentence must be true; but if this statement is true then the sentence must be false. And so we go, around and around, a never ending back-and-forth where each truth-value interpretation of the sentence could be said to ‘vanish into its opposite’. This self-referential sentence lacks a determinate truth-value.

But assigning truth values, or any kind of values, to pure being and pure nothing, cannot be done. This would introduce ‘determinations’ that don’t yet exist in the phenomena. So far, we merely have a metaphysical bedrock that seems to support two opposite concepts, which are nonetheless intimately related. Each concept immediately implies the other, and therefore cannot exist alone. To try to make sense of this, let’s re-examine pure being and pure nothing.

Yes versus no

Hegel says pure being and pure nothing are the same and yet also ‘not the same, that they are absolutely distinct’. Let’s focus first on their difference.

Being has a positive aspect, or connotation, of existence, of something that is present. In contrast, nothing has the negative aspect of non-existence, of something that is absent. This is the clear distinction between them: being is (in some obscure sense) positive, and nothing is (in an equally obscure sense) negative. So let’s draw our diagrams again but this time include those distinctions:

IMG_20181003_144841
Pure being and pure nothing are distinct: being implies presence (+), nothing implies absence (-).

So now we have a distinction between pure being and pure nothing. But we also have to comprehend why each ‘vanishes into its opposite’ and why – despite their difference – they are in fact the same. Let’s consider the vanishing aspect first.

In the Epimenides paradox we try to evaluate the truth value of ‘This sentence is false’ and, in doing so, flip back-and-forth between true and false. In our hands, the true sentence morphs into a false sentence, and vice versa. But Hegel here talks of pure being (or pure nothing) vanishing into its opposite. There is no ‘I’ or ‘we’, or metaphorical hands, to perform this vanishing trick; instead, according to Hegel, it is pure being (or pure nothing) itself that vanishes into its opposite, as an active agent. In other words, pure being (or pure nothing) actively alters, or changes, into its opposite. As far as I can tell, Hegel’s beginning is irreducibly dynamic in the sense that the core concepts – pure being and pure nothing – are not merely concepts but actual occurrent processes (that we intuit by self-referential philosophical reflection).

Vanishing implies that pure being (and pure nothing) may alter or change in such a way to become their opposite. Here, at this point, I will begin to apply the mathematical language of the calculus to represent change. Note that I don’t claim that the calculus is right there at the metaphysical bedrock, just as Hegel does not claim that natural language (in fact, German and Latin) are there. Rather, we simply have no choice but to use currently available languages to describe the phenomena, but, of course, the phenomena isn’t those languages. The calculus merely serves as a useful language to attempt to describe and understand the bedrock.

(In fact, both Hegel and Marx were intensely interested in the mathematical calculus of their day, and both wrote extensively about it. Hegel discusses the calculus later in his Logic, but we’ll postpone investigating what he said about the calculus for another time).

Now, change implies a sequence of different ‘states’ that the thing that changes exhibits. So we need some way to talk about and denote the elements of the sequence. Let’s use subscript ‘t‘ to index elements of a sequence. (But let’s not over interpret and start thinking about the existence of time by this choice; rather, it’s simply a convenient reminder that the sequences of states are ordered).

So, in order to vanish, pure being must change in some way, which we shall describe in terms of the following incomplete differential equation:

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Pure being changes, but at the moment we’re not sure how

But how does it change? Well, in whatever way it does change, that change must reflect the distinctiveness of pure being from pure nothing. And, as we’ve already stated, pure being is distinct from pure nothing because it has the positive aspect of existence: it’s a kind of affirmation of existence, a great ‘yes!’ So let’s add this distinction to the kind of change pure being manifests:

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Pure being has a positive aspect, which is what distinguishes it from pure nothing

Here we’re borrowing the “positiveness” of an arithmetic operator to represent the “positiveness” of pure being. That may be an inadequate representation, but no more inadequate than the corresponding natural language term. So we state, in the above incomplete differential equation, that pure being changes in a positive or affirmative manner: it has a distinct direction of change.

But what can pure being change with respect to? What causes it to change? The answer here, is very clear, it can only change with respect to itself. As Hegel says, here at the beginning, ‘Nor is there anything else present, any content which could be used to make the beginning more determinate’. So we must complete the specification of the change that pure being must manifest as follows:

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Pure being is everything and therefore can only change in terms of itself

In other words, pure being changes positively with respect to itself.

Let’s examine the same issue from the perspective of pure nothing. Again, the change that pure being can manifest must reflect its distinctiveness from pure being. Pure nothing is distinct from pure being because it has the negative aspect of existence: it’s a denial of existence, a great ‘no!’ And, in an identical manner, pure nothing can only change with respect to itself, for pure nothing necessarily implies there is nothing else to change with respect to. In consequence, for symmetrical reasons we must specify the change that pure nothing manifests as the following differential equation:

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Pure nothing has a negative aspect, which distinguishes it from pure being, and it can only change in terms of itself

So we now have pure being and pure nothing, as we always had, but with some additional structure that describes how pure being and pure nothing must change (which is a prerequisite to understanding how they might vanish via their own activity).

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Pure being and pure nothing as “absolutely distinct” starting points, which lack any content, yet may change with respect to themselves.

Coming-to-be and ceasing-to-be

Quick reminder where we are: Hegel claims that pure being and pure nothing are ‘absolutely distinct’ yet are ‘unseparated and inseparable’ and somehow ‘vanish’ into each other and therefore are, in fact, ‘the same’. In the above diagram we’ve reconstructed only some of these claims: pure being and pure nothing are absolutely distinct (due to one being positive and the other negative) and have the capability to change with respect to themselves. Being able to change is a precondition of being able to vanish. So do they?

Let’s solve the ‘equation of motion’ of pure being. It’s one of the simplest possible differential equations, but for those who may be a little rusty, I include the full derivation:

IMG_20181004_115659
Pure being changes positively with respect to itself; in consequence, pure being increases exponentially.

Now that we’ve solved the differential equation we can plot the ‘trajectory’ of pure being (i.e., its sequence of states indexed by ‘t‘). We see that it increases exponentially and approaches infinity (from whatever arbitrary starting point we might specify):

beingChange
Pure being is ‘coming-to-be’ as it speeds towards infinity

On the face of it pure being is definitely not vanishing! It’s getting ‘bigger’!

But Hegel doesn’t claim that pure being merely vanishes, but rather that pure being vanishes into its opposite. However, that doesn’t seem to be happening either, since, if we were tempted to associate numerical concepts to pure being and pure nothing, then an exponential approach to infinity isn’t ‘nothing’.

But before we rush to judgement, let’s examine the behaviour of pure nothing by solving its (different) equation of motion:

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Pure nothing changes negatively with respect to itself; in consequence, pure nothing decreases exponentially

The negativity of pure nothing makes a big difference: from whatever arbitrary starting point pure nothing exponentially decreases to zero:

nothingChange
Pure nothing is ‘ceasing-to-be’ as it speeds towards zero

On the face of it pure nothing is vanishing: it’s getting ‘smaller’. But, again, Hegel does not claim that pure nothing merely vanishes, but rather it vanishes into its opposite. So pure nothing isn’t vanishing in the way Hegel intends. And doesn’t seem to be vanishing into its opposite, which is pure being.

Let’s examine the matter a little more closely.

Why pure being and pure nothing are the same

Hegel’s ‘vanishing into its opposite’ is similar to the Epimenides paradox where the self-referential sentence flips between truth values. But the vanishing isn’t only ‘logical’ since pure being and pure nothing change into their opposites. So in some sense there is an additional ‘causal’ aspect to the vanishing (although cause and effect have no place here). By considering how pure being and pure nothing necessarily self-relate we’ve seen that pure being is a coming-to-be and pure nothing is a ceasing-to-be, but, as yet, we haven’t seen either of these purities vanishing into its opposite.

But we can see this happening, in multiple senses, if we look carefully.

The language of the calculus may help but it can also hinder. It can hinder because it’s loaded with semantics that we don’t wish to adopt, at least not yet. For example, change has numerical value, but – at this assumption free stage of analysis – we don’t have the concept of magnitude or number, we literally have only the phenomena of pure being and pure nothing. So the fact that pure being and pure nothing are self-referential and distinct, and can change, should not imply they either increase or decrease in ‘size’, or even that the change takes place in ‘time’. So we need to shed some of the unwanted semantics of the calculus at this point. (Indeed, Hegel also sheds the additional semantic baggage of ordinary language terms when describing the bedrock, and coins various neologisms and makes extensive remarks and commentary to the main argument, in an attempt to clarify.)

We can avoid the unwanted semantics by looking at the ‘shape’ of the trajectories of pure being and nothing – and noticing that those shapes are identical:

flip
The coming-to-be of pure being and the ceasing-to-be of pure nothing are identical once we remove the idea of an arrow of time. Pure being approaches ∞ as t increases (top-left). But equally pure being approaches 0 as t decreases (top-right). Pure nothing supports a similar reversal of perspective (bottom-left and bottom-right). We can therefore see that pure being is identical to (reversed) pure nothing, and pure nothing is identical to (reversed) pure being. This is a simple consequence of the fact that x(-t)=y(t) and y(-t)=x(t), when x(0)=y(0).

So, although pure being and pure nothing are distinguished by their positive and negative aspects, we can see that their behaviour, with respect to themselves, is exactly the same (once we shed the unwarranted assumption that ‘t‘ denotes time that moves forward).

We can make this sameness more precise in terms of a reciprocal map between the trajectories of pure being and pure nothing:

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The reciprocal map, f(x)=1/x, defines a mapping between the behaviour of pure being and pure nothing (and vice versa). Pure being and pure nothing are therefore isomorphic to each other. (N.B. the arrows in the above diagram don’t represent self-reference, but the mapping operation).

Pictorially, the situation is:

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Pure being and pure nothing have distinct behaviours with respect to themselves (trajectories), denoted x(t) and y(t), respectively. The behaviour of pure being has a one-one relationship with the behaviour of pure nothing via the reciprocal map, f(x)=1/x. Since the trajectories are isomorphic, pure being and pure nothing are ‘the same’.

So we can make sense of Hegel’s startling claim that pure being and pure nothing are ‘the same’ and also ‘they are not the same’. They are not the same because pure being is about existence, whereas pure nothing is about non-existence, and hence they self-interact in different ways (pure being affirms itself, pure nothing denies itself). But they are also the same because the shape of these self-interactions are isomorphic. Their behaviour is identical.

The fact that so far we’ve been dealing with two concepts – pure being and pure nothing – arose initially because pure being implies pure nothing, and pure nothing implies pure being. So, like the Epimenides paradox, when we contemplate the metaphysical bedrock we are forced to flip back-and-forth between two different, equally immediate, fundamental viewpoints. In this, ‘logical’ sense, pure being and pure nothing also vanish into each other.

Furthermore, when we observe how pure being and pure nothing necessarily self-relate in distinctive ways we notice two different behaviours, the ‘coming-to-be’ of pure being and the ‘ceasing-to-be’ of pure nothing. In the language of the calculus the trajectory of pure being is an exponential increase to infinity, whereas the trajectory of pure nothing is an exponential decrease to zero.

But these different behaviours, as we’ve just seen, are isomorphic to each other, via the reciprocal map, f(x)=1/x. Notably, the map is an involution, i.e. f(f(x))=x, and therefore is its own inverse. So we can flip back-and-forth between the distinct behaviours of pure being and pure nothing, by applying the involution, forever. The structure of the Epimenides paradox therefore also holds when we think of pure being and pure nothing as coming-to-be and ceasing-to-be. And therefore they vanish into each other in this fuller sense too.

But Hegel’s kind of vanishing isn’t merely ‘logical’ but seems to have a ‘causal’ aspect to it. Can we make sense of the idea that pure being and pure nothing themselves vanish into their opposites?

The ‘incomprehensibility of the beginning’

Hegel also talks of a ‘movement of the immediate vanishing’ such that pure being and pure nothing ‘vanish into its opposite’, an occurrence he labels becoming, which is this endless back-and-forth between pure being and pure nothing:

… becoming is the vanishing of being into nothing, and of nothing into being, and the vanishing of being and nothing in general; but at the same time it rests on their being distinct.

So all this vanishing is also ‘the vanishing of being and nothing in general’. Everything vanishes!

In the language of calculus, the coming-to-be of pure being speeds towards infinity. In this sense, pure being explodes. For example, any natural or mechanical systems (which we cannot properly talk about here, but only mention by analogy) undergoing exponential growth quickly fall apart. They could only exist, at best, for a short period of time. Similarly, the ceasing-to-be of pure nothing collapses to zero. Again, any natural or mechanical systems that obeyed this exponential law of decrease would quickly cease and become entirely inert. In this sense, pure nothing implodes. In both cases, the systems ‘vanish’, either by exploding or imploding. So although pure being and pure nothing are present at the metaphysical bedrock, and imply each other, as self-referential systems they are unstable. They cannot permanently exist in their pure states of self-reflection.

The asymptote of coming-to-be is infinity and we could therefore say, in the language of the calculus, that pure being is the coming-to-be of infinity. But pure being will never reach that state (even exponential growth does not reach infinity). Thought contemplating itself can never catch its own tail, but will endlessly chase it, caught forever in a self-referential loop.

Similarly, the asymptote of ceasing-to-be is zero and we could also say that pure nothing is the ceasing-to-be towards zero or nullity. But pure nothing will never reach that state (even exponential decrease never gets to zero). As thought contemplating nothing it can never eradicate its own existence, and therefore forever sustains some residual of thought in the self-referential loop.

In both cases, coming-to-be and ceasing-to-be are incomplete, and never reach their fully pure terminal states (of infinity or zero). I’m tempted to slightly revise Hegel’s terminology and reserve the terms pure being and pure nothing for their asymptotic states of infinity and zero, respectively. We’d then use the terms being and nothing to refer to the distinct self-referential processes (which vainly strive for their pure states, but are tragically condemned to never reach them). So, in this specific sense, pure being and pure nothing cannot exist.

We can now, at last, explicate Hegel’s ‘causal’ sense of vanishing into an opposite. Consider pure being in the state epsilon close to zero (where we imagine epsilon is a really small magnitude as close to zero as we wish). This state is arbitrarily close to the asymptote of pure nothing. But since being is a coming-to-be this ’empty’ state rapidly vanishes towards the state of pure being (infinity). More prosaically: when we try to contemplate absolutely nothing there’s an irreducible element of our own existence, which when noticed, takes over, and flowers into pure being.

beingChange
Being: any state arbitrarily close to pure nothing (zero) vanishes into its opposite, that is pure being (infinity)

Similarly, consider nothing in a state close to pure being (i.e., 1/epsilon ‘close’ to infinity, where epsilon is a really small magnitude). Since nothing is a ceasing-to-be this ‘full’ state rapidly vanishes towards the state of pure nothing (zero). And, once again, in very prosaic terms: when we try to contemplate that which is common to everything, or pure existence itself, there’s an irreducible element of non-existence or nothing, which when noticed, takes over, and decays into pure nothing.

nothingChange
Nothing: any state arbitrarily close to pure being (infinity) vanishes into its opposite, that is pure nothing (zero)

So, after a surprising amount of work, I think we can now make perfect sense of Hegel’s surprising paragraph, which I re-quote here:

Pure Being and pure nothing are, therefore, the same. What is the truth is neither being nor nothing, but that being β€” does not pass over but has passed over β€” into nothing, and nothing into being. But it is equally true that they are not undistinguished from each other, that, on the contrary, they are not the same, that they are absolutely distinct, and yet that they are unseparated and inseparable and that each immediately vanishes in its opposite. Their truth is therefore, this movement of the immediate vanishing of the one into the other: becoming, a movement in which both are distinguished, but by a difference which has equally immediately resolved itself.

We can make sense of Hegel’s beginning by interpreting this passage as essentially talking about isomorphic positive and negative feedback loops. Hegel’s philosophical reflection asks us to perform the following mental exercise: shed all your knowledge and assumptions, including your own existence as an individual person and simply contemplate the existence of thought. When you do that you’ll enter a self-referential feedback loop. This mental state has paradoxical properties (especially from the point-of-view of formal logic). First, it seems like you are contemplating pure being (that is existence itself) but also it seems like you are contemplating pure nothing (zero content, or non-existence). Both points-of-view make sense. But the interpretations are unstable, and don’t settle down, and spontaneously flip back-and-forth. And, furthermore, what the feedback loops seem to strive towards – the state of pure being or pure nothing – can never be reached. In this sense, they cannot exist, both ‘logically’ and ‘causally’.

We might expect that clear distinctions between logical and causal necessity shatter on the metaphysical bedrock.

‘Becoming’, this ceaseless flip-flop between being and nothing, implies ‘the vanishing of being and nothing in general’. So the beginning is paradoxical, and as understood so far, seems also to be both ‘logically’ impossible and ‘causally’ unstable. This beginning doesn’t make sense and it cannot exist. As Hegel intimates, the beginning appears, at first, to be incomprehensible.

Is the whole endeavour a non-starter? How does Hegel resolve this? This is where Hegel begins to get really interesting.

The sublation of being and nothing

Since the beginning doesn’t make sense and cannot exist it therefore cannot really be the beginning after all. The beginning must be something else. At this point Hegel introduces the concept of ‘sublation’:

Becoming is the unseparatedness of being and nothing, not the unity that abstracts from being and nothing; as the unity of being and nothing it is rather this determinate unity, or one in which being and nothing equally are. However, inasmuch as being and nothing are each unseparated from its other, each is not. In this unity, therefore, they are, but as vanishing, only as sublated. They sink from their initially represented self-subsistence into moments which are still distinguished but at the same time sublated.

Let’s take this line by line. Hegel states, ‘Becoming is the unseparatedness of being and nothing, not the unity that abstracts from being and nothing’. Recall earlier that I introduced the reciprocal map to abstract from the difference between being and nothing, and point out they are isomorphic to one another. But Hegel now says that ‘becoming’ is not such an abstraction of being and nothing but their ‘unseraparatedness’. So our original conception of the relationship between being and nothing – which doesn’t make any sense and cannot exist – wasn’t right.

Instead, becoming must be ‘the unity of being and nothing … in which being and nothing equally are’. So somehow we need to think of being and nothing as joined in a unity (‘unseperated from its other’) and in this unity they are ‘vanishing’ but in a sublated manner such that ‘they sink from their initially represented self-subsistence’, i.e. we no longer think of being and nothing as relating only to themselves, ‘into moments which are still distinguished but at the same time sublated’.

How might we join being and nothing in a unity?

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Pure being and pure nothing as purely self-relating concepts don’t make sense and cannot exist. So they must exist together as a “sublated unity”. What could that mean?

Of course, the simplest way of joining these feedback systems together is to literally join them together: the output of being becomes the input of nothing, and the output of nothing becomes the input of being:

IMG_20181009_163234
Becoming is the sublated unity of being and nothing. They no longer self-relate but ‘interpenetrate’ each other. This is the most important picture in this post.

In this unity, being is no longer affirming its own existence but now is affirming non-existence, or nothing (in the language of calculus, being now positively changes with respect to nothing, dx/dt=y). On the other hand, nothing is no longer negating its own existence but now is negating being (nothing now negatively changes with respect to being, dy/dt=-x). Poetically, being acknowledges the existence of nothing, and nothing denies the existence of being.

We do seem to have captured some aspects of Hegel’s sublated unity by constructing a system of coupled differential equations, since being and nothing are now joined together, but they remain distinct.

img_20181011_142033.jpg
The unity of being and nothing expressed as a system of coupled first-order differential equations. We need to refer to these equations. I’ll call them ‘Hegel’s equations‘, and sometimes ‘Hegel’s contradiction‘.

Hegel continues:

Grasped as thus distinguished, each is in their distinguishedness a unity with the other. Becoming thus contains being and nothing as two such unities, each of which is itself unity of being and nothing; the one is being as immediate and as reference to nothing; the other is nothing as immediate and as reference to being; in these unities the determinations are of unequal value.

In the language of the calculus, becoming does indeed now contain being and nothing as ‘two such unities’ where ‘one is being as immediate and as reference to nothing’ (if we start with being in the above figure then its input refers to nothing) and ‘the other is nothing as immediate and as reference to being’ (and if we start with nothing then its input does refers to being). And, furthermore, the ‘determinations are of unequal value’ in the straightforward sense that, in general, dx/dt is not equal to dy/dt.

So the coupled system mirrors Hegel’s natural language description of sublation remarkably well. Hegel continues:

Becoming is in this way doubly determined. In one determination, nothing is the immediate, that is, the determination begins with nothing and this refers to being; that is to say, it passes over into it. In the other determination, being is the immediate, that is, the determination begins with being and this passes over into nothing – coming-to-be and ceasing-to-be.

Hegel here repeats that being and nothing no longer self-relate but refer to each other. But this isn’t simple logical reference but has a causal aspect too: the reference is such ‘that is to say, it passes over into it‘. And, in the coupled system, we have a ‘substance’ that actually flows from being into nothing, and the ‘substance’ leaves nothing and enters into being. The previous positive (coming-to-be) and negative (ceasing-to-be) aspects of being and nothing are also in the coupled system: nothing still negates and implodes (i.e. reacts negatively to its input) and being still affirms and explodes (i.e., reacts positively to its input).

Hegel then closes his description of the ‘sublation of being and nothing’ with the following:

Both are the same, becoming, and even as directions that are so different they interpenetrate and paralyze each other. The one is ceasing-to-be; being passes over into nothing, but nothing is just as much the opposite of itself, the passing-over into being, coming-to-be. This coming-to-be is the other direction; nothing goes over into being, but being equally sublates itself and is rather the passing-over into nothing; it is ceasing-to-be. – They do not sublate themselves reciprocally – the one sublating the other externally – but each rather sublates itself in itself and is within it the opposite of itself.

In Hegel’s equations being and nothing ‘interpenetrate’ each other; but we have yet to see whether they ‘paralyze’ each other. In the language of the calculus we can interpret Hegel’s clarification that ‘they do not sublate themselves reciprocally’ or ‘externally – but each rather sublates itself in itself and is within it the opposite of itself’ to imply that being takes nothing as input and nothing takes being as input so each is ‘within it the opposite of itself’.

Pre sublation being and nothing were separate concepts that didn’t make sense and could not exist. Post sublation we have a unity of being and nothing – they are joined, and interact with each other . This unity is contradictory in the sense that being affirms whereas nothing negates.

So, does our more refined concept of the beginning – as the sublated unity of being and nothing – now make sense? and can it exist?

Becoming

Hegel now introduces the idea of an equilibrium in another difficult paragraph:

The equilibrium in which coming-to-be and ceasing-to-be are poised is in the first place becoming itself. But this becoming equally collects itself in quiescent unity. Being and nothing are in it only as vanishing; becoming itself, however, is only by virtue of their being distinguished. Their vanishing is therefore the vanishing of becoming, or the vanishing of the vanishing itself. Becoming is a ceaseless unrest that collapses into a quiescent result.

Hegel’s equations do indeed state that ‘being and nothing are in it only as vanishing’ in the sense that being is ‘vanished’ by its negation by nothing (negative feedback) and nothing is ‘vanished’ by its affirmation by being (positive feedback).

But now we also have a typically Hegelian claim: becoming is both a ‘ceaseless unrest’ and a ‘quiescent result’. So the ‘vanishing’ that previously implied that pure being and pure nothing could not exist, now, in this sublated state, ‘vanishes the vanishing itself’ such that we now have ceaseless unrest that paradoxically collapses into a stable result (that presumably doesn’t vanish by either exploding or imploding).

Does it? An advantage of the language of the calculus, in contrast to Hegel’s natural language, is that we can apply the formal machinery of mathematics to actually check whether Hegel’s unity, as he describes it, in fact generates a ‘ceaseless unrest’ that ‘collapses into a quiescent result’. We can solve Hegel’s equations to derive the ‘equations of motion’ of being and nothing in their sublated state. What kind of new ‘vanishing’ do we find?

IMG_20181009_174735
Solving Hegel’s unity of being (x) and nothing (y). First we notice that the second derivatives of x and y are independent, and, at this level of description, are the same.

We haven’t solved Hegel’s equations yet. But the above deduction reveals that Hegel’s equations are equivalent to two second-order differential equations that, in terms of their motion, are independent (although they are coupled in terms of their initial conditions, as we shall see shortly).

The equations for being and nothing, in this form, are identical. They are clearly isomorphic at this level of description. Also, each ‘pole’ of the unity changes with respect to itself and therefore being and nothing are still self-relating, but in a different, sublated form: now we have the acceleration of being (or nothing) changing negatively with respect to itself.

Next, we need to solve the second-order linear differential equations. Since this deduction applies to both being (x) and nothing (y) I’ll just solve for the temporary variable (z):

img_20181010_172621.jpg
Standard solution of a linear second-order differential equation.

So being and nothing exhibit sinusoidal or oscillatory motion. But we haven’t quite solved Hegel’s equations quite yet. We must remember that Hegel’s equations are coupled and therefore observe a constraint between each other. There is an essential duality within becoming that we mustn’t neglect.

To completely solve Hegel’s equations we must specify initial conditions for being and nothing. These values are completely arbitrary, except we must ensure that being and nothing preserve their distinctiveness in their sublated unity. So we stipulate that being, when t=0, has a positive value (to denote its positive existence) that, without loss of generality, we will set to 1, i.e. x(0)=1. And we’ll stipulate that nothing starts at the different value of zero (to denote its non-existence), i.e. y(0)=0. Let’s use these two differing starting conditions to completely solve Hegel’s equations:

IMG_20181011_124618
The solution to Hegel’s equations, which describe how being and nothing change when sublated as a process of becoming.

So, in this mathematical interpretation of becoming, being changes according to cos(t) and nothing changes according to -sin(t). Now we are in a position to check Hegel’s claim that becoming is a ‘ceaseless unrest’ that ‘collapses into a quiescent result’.

Becoming as a ‘ceaseless unrest’

Below I plot the solution to Hegel’s equations, which shows how being and nothing change over ‘time’:

becomingPlot
Becoming as ‘ceaseless unrest’: being and nothing do not settle to a fixed value, but continually come-to-be and cease-to-be in virtue of their interaction with each other.

The plot shows that being and nothing oscillate between -1 and 1, forever. They oscillate identically (since both their motions, as we saw, are governed by the same second-order differential equation) but they are permanently out of phase. When being realises its maximum (at an absolute value of 1) then nothing is at its absolute minimum of 0. When nothing realises its maximum, then being is at its minimum. What one ‘gains’ the other ‘loses’, but neither ‘side’ ever wins.

What is being gained, and what is being lost? It’s tempting to introduce familiar physics-based concepts, such as amplitude or energy etc. But Hegel employs the term ‘indeterminate being’, which is the status of pure being and pure nothing prior to their sublation. I will use the slightly more evocative term ‘substance’. Being and nothing continually exchange their substance with one another: at one time being is more substantial, at another time nothing is. Define the total substance contained within the unity as the sum of the squares of x and y (to handle the negative values). As soon as we do that, we immediately see that Hegel’s equations instantiate a simple conservation law:

IMG_20181012_134512
Conservation of ‘substance’: as being and nothing take turns to wax and wane the total substance within the unity is conserved.

So becoming is a process where coming-to-be (being) negates ceasing-to-be (nothing), and ceasing-to-be (nothing) negates coming-to-be (being) – forever. The substance of being and nothing each ‘vanish’ into each other (and continually reappears); or, as Hegel states:

… becoming is the vanishing of being into nothing, and of nothing into being, and the vanishing of being and nothing in general; but at the same time it rests on their being distinct.

We can think of this perpetual trade-off between being and nothing as either eternal conflict, or an eternal dance of co-operation. Hegel, more simply, describes it as ‘a ceaseless unrest’. The union of being and nothing is unstable, any equilibrium is immediately undermined, and the opposing concepts remain in perpetual contradiction. As Hegel describes it:

It therefore contradicts itself in itself, because what it unites within itself is self-opposed; but such a union destroys itself.

So we’ve shown that Hegel’s contradiction does generate ceaseless unrest. But in typically Hegelian fashion, becoming is not merely a ‘ceaseless unrest’ but also a ‘quiescent result’. Can we make sense of this too?

Becoming as a ‘quiescent result’

Every moment of becoming is characterised by a pair of values, x(t) and y(t). Each pair belongs together, and define the instantaneous state of becoming. The set of every such pair, (x(t), y(t)), defines the state-space of becoming, which we now plot:

unity
Becoming as a ‘quiescent result’: being and nothing always vary but are bounded. The 2-D state-space of Hegel’s equations is a perfect circle.

In other words, Hegel’s equations, that is the unity of being and nothing, trace a perfect circle in state-space. Becoming is indeed ceaseless unrest, but that unrest is always bounded. Pure being, which merely self-relates, explodes, and pure nothing, which also self-relates, implodes; in this sense, neither can exist. In contrast, Hegel’s equations define a stable dynamic system: their sublated unity neither explodes or implodes, but is a ‘quiescent result’ that reproduces itself indefinitely.

In fact if we – somehow – managed to be outside observers and ‘measured’ the total substance of becoming we would notice no change whatsoever. Hegel’s equations, as we’ve seen, obey a conservation law. The ceaseless unrest on the inside conserves the total substance and so, on the outside, we would observe perfect calm, a truly quiescent result. So becoming both preserves its identity over time (conservation of substance) and changes (the internal oscillation).

Becoming exhibits a new kind of ‘vanishing’ different to the ‘vanishing’ we originally observed when the concepts were purely self-relating (which Hegel refers to as the ‘already sublated determinations’ below). But as Hegel notes, although vanishing is preserved, it is also changes, within the sublated unity:

This result is a vanishedness, but it is not nothing; as such, it would be only a relapse into one of the already sublated determinations and not the result of nothing and of being. It is the unity of being and nothing that has become quiescent simplicity. But this quiescent simplicity is being, yet no longer for itself but as determination of the whole.

The unity of being and nothing determines a new kind of whole: a dynamic and contradictory unity. Pure being and nothing ‘sink from their initially represented self-subsistence’ and are turned into ‘moments’ of a bigger whole where they are ‘distinguished but at the same time sublated’.

Existence: its necessary metaphysical structure

To recap: we saw that pure being, as a self-referencing concept, was logically unstable since it implied another concept, pure nothing. These concepts were meant to refer only to themselves, yet they implied an opposite concept, which was the same and also different. And so this beginning didn’t make logical sense. Also, since there is no ‘I’ in the assumption-free and indeterminate beginning, the movement or vanishing of these bedrock concepts must be their own self activity: it is they that ‘vanish’ into each other, and not us (or any other agency) that makes them do it. From this more ‘causal’ point-of-view, pure being and pure nothing are also the same but different, and furthermore, necessarily ‘vanish’ in the sense of exploding or imploding. We concluded, following Hegel, that this beginning doesn’t make sense and cannot exist. The beginning is a paradox.

Hegel resolves this paradox by a (logical? causal?) operator he calls sublation. Hegel remarks that sublation is ‘one of the most important notions in philosophy’. A sublation, in typically Hegelian fashion, both preserves or maintains and puts an end to. Where did this operator come from? I think Hegel would argue that this operator is observable within the phenomenon itself. By observing pure being and pure nothing, with no assumptions whatsoever, we learn that: being and nothing are the same, and yet they are different; and that they imply each other and must exist together. So they cannot exist as purely self-referential concepts. The only possible way they can exist together is as some kind of higher unity. We initially observed this higher unity in only a ‘one-sided’ or limited manner. The beginning was a paradox because we thought we were observing two independent objects but really they were aspects of just one.

Becoming is the name Hegel gives to the sublation of pure being and pure nothing. Suddenly, everything changes: we ‘put an end to’ pure being and pure nothing as self-referential concepts (as uncoupled differential equations); and now they reciprocally refer to each other (as coupled differential equations). So each presupposes the other, and neither is a unique starting point. There cannot be being without nothing, or nothing without being. The logical paradox is resolved.

The sublation preserves the distinctiveness of being and nothing: Hegel’s equations are two different, first-order differential equations, where being has a positive aspect, and nothing has a negative aspect. But also their sameness is also preserved: Hegel’s equations may also be written as two identical, second-order differential equations, and therefore they are the same: the moments they trace out are phase shifted but identical.

Furthermore, sublation preserves the ‘vanishing’ of being and nothing. But instead of exploding or imploding, they now ‘interpenetrate each other’ and mutually ‘vanish’ into each other by exchanging their substance in an oscillatory but conservative manner. Becoming is therefore a ceaseless unrest that nonetheless remains stable over time. The causal paradox is resolved.

In consequence, the unity of being and nothing determines a beginning that does make sense and can exist: the beginning is an irreducibly dynamic and contradictory unity.

According to Hegel the fundamental structure of becoming must be present, as both a logical and natural necessity, in anything that exists at all. Hegel states: ‘there is nothing which is not an intermediate state between being and nothing’. What is particularly startling and interesting about Hegel’s metaphysical argument is that it implies that negativity, nothing or non-existence is not the absence of being but a necessary and irreducible kind of being. Hegel is a substance monist but his substance has two fundamentally different aspects.

Hegel closes the first chapter of Science of Logic by stating that ‘existence’ is necessarily the unity of being and nothing:

Becoming, as transition into the unity of being and nothing, a unity which is as existent or has the shape of the one-sided immediate unity of these moments, is existence.

And I’ll close my mathematical interpretation of Hegel’s first chapter with the following table:

grid

A mathematical interpretation of the first chapter of Hegel’s Science of Logic

Existence: harmonic oscillation all the way up, and all the way down

But so what? Who cares? Is there any evidence that everything that exists has this fundamental structure? Or is this merely – although admittedly beautiful – metaphysical and mathematical poetry?

My mathematical interpretation covers only chapter 1 of Hegel’s monumental and obscure Science of Logic. In subsequent chapters, Hegel derives the necessary existence of further categories, such as quality, finitude, infinity, multiplicity, quantity, measure and the syllogisms of ‘ordinary’ logic. We should explore how far this new, mathematical interpretation of Hegel’s opening chapter extends to his later chapters. At some point, the semantics of Hegel’s metaphysical theory and the semantics of systems of differential equations must surely break down. But who knows? We might yield more insights into Hegel’s philosophy by pursuing this project. Regardless, Hegel – at least as far as he is concerned – derives and critiques the Kantian categories from his assumption free starting point, and, if this derivation is successful, then that would constitute evidence that fundamental aspects of our cognition are the manifestation of the contradiction between being and nothing.

What about physical reality?

Let’s return to Hegel’s equations represented as second-order differential equations. We have:

IMG_20181013_104312-cropped
Hegel’s equations written as a pair of identical, second-order differential equations. Each equation is identical to a simple harmonic oscillator.

 

Those with a physics background will have already noticed that Hegel’s equations imply that the unity of being and nothing instantiates simple harmonic oscillators. Simple harmonic oscillators are the bread-and-butter of physics courses simply because harmonic oscillation is ubiquitous in nature, both in the microcosm (quantum) and the macrocosm (general relativity). As above, so below. Quantum field theory, the currently dominant theory of fundamental particles, is essentially simple harmonic motion taken to increasing levels of abstraction. In other words, simple harmonic motion is indeed a fundamental structure that appears, again and again, at all levels of physical reality.

I plan to return and expand upon this point, especially as Hegel’s contradiction is not merely simple harmonic motion, but rather a 2-D, system of coupled harmonic oscillators with additional properties that relate to complex analysis and holomorphic functions. But here let’s simply note the following: it’s utterly remarkable that Hegel’s psychedelic, assumption-free starting point, which is resolutely conceptual and abstract – and makes no reference to physical reality or empirical knowledge whatsoever – nonetheless, according to the interpretation developed here, implies a structure of ‘becoming’ that is equivalent to the fundamental structure found everywhere in physical reality.

This result has renewed and reinvigorated my interest in Hegel, and I hope it has the same effect on you.

Physicists, over the centuries, have observed and interacted with empirical reality and developed theories that feature harmonic oscillation. In this sense, they have described the world as they have found it. But physicists might not ask, and perhaps could not answer, why oscillatory motion is ubiquitous in nature. Philosophy, in particular Hegel’s metaphysics, in contrast, provides a candidate explanation of this empirical phenomenon: according to Hegel, everything that exists necessarily is a unity of being and nothing and therefore – according to the mathematical interpretation developed here – must exhibit harmonic motion.

In a subsequent post I will explain how Hegel’s contradiction also manifests in number theory. One would think that the natural numbers are irredeemably static and impervious to change. But we’ll see that Hegel’s concept of a dynamic, contradictory unity already appears in this field of mathematics, although conceptualised without reference to Hegelian metaphysics.

Recommended reading

Hegel’s Science of Logic

Stephen Houlgate‘s The Opening of Hegel’s Logic, published in 2006, is the best secondary source I have read on Hegel’s Logic. (Watch out, I’ve tried quite a few commentaries, and most are very bad).

Evald Ilyenkov‘s 1970s book, Dialectical Logic, is a fantastic historical account of the relevance of German Idealism to Marxist materialism. Ilyenkov sadly committed suicide in 1979.

 

7 Comments

  1. Are you familiar with category theory (in mathematics)? One of its developers (I forget which one) was greatly influenced by Hegel.

    It’s a very useful conceptual tool. David I. Spivak (at MIT) has written an accessible book on it for a wide audience (including economists) titled β€œCategory Theory for Scientists”, he put a old pdf version of it online:

    Click to access CT4S.pdf

    Speaking of mathematics, what do you think of mathematizations of Marx offered by Ulrich Krause in his book β€œMoney and Abstract Labour: On the Analytical Foundations of Political Economy” of and by Emmanuel Farjoun and MoshΓ© Machover in their book β€œLaws of Chaos: A Probabilistic Approach to
    Political Economy”?

    Liked by 1 person

  2. Hi Penny

    Thanks for your comment!

    You’re probably thinking of William Lawvere:

    https://en.wikipedia.org/wiki/William_Lawvere

    who made connections between dialectical materialism and category theory. Sadly I haven’t studied category theory, but would like to understand more, so I have an unread copy of Lawvere’s “Conceptual Mathematics” introductory textbook on category theory. My slight worry, however, is that category theory is like a very high-level programming language: it’s fun to rewrite code very succinctly, with a small number of reusable concepts, but fundamentally you don’t do anything new: it’s just an elegant re-description.

    The n-Lab pages have interesting, if somewhat obscure (to me at least), links between higher mathematics and dialectics; e.g. here’s their page on Lawvere

    https://ncatlab.org/nlab/show/William+Lawvere

    I think a litmus test for Hegelian dialectics is that it yield novel perspectives on foundational issues in philosophy, mathematics and physics. This follows from the logic of Hegel’s own argument. For if Hegel’s derivations really do have the force of necessity then the “Hegelian contradiction” should appear in all these fields. That we get oscillatory/wave-like behaviour from Hegel’s derivation is a great start. But for me this remains an interesting open question.

    I have a great deal of respect for the work of Ulrich Krause, and I studied his book “Money and Abstract Labour” quite closely some years ago. Krause recasts the transformation problem in the context of heterogeneous productivity of labour. Although closer to empirical reality I think this approach avoids, rather than gets to the root, of the problematic of the relationship between money and labour time. For example, the transformation problem arises even in “ideal conditions” for a labour theory of value, such as homogeneous productivity of labour. A lot more to say of course, but I just wanted to share my general attitude. Certainly, Krause’s book is well worth reading for a broader understanding of the mathematical structure of the labour theory of value.

    Thanks for the link to the category theory PDF. I’m on the cusp of deciding to delve in, or avoid altogether. Unsure!

    Best wishes,
    -Ian.

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    1. I forgot to answer your query about Farjoun & Machover’s “Laws of Chaos”. I think “Laws of Chaos” is the most significant contribution to Marxist political economy since I. I. Rubin’s “Essays on Marx’s Theory of Value”. In 2008 I helped organise a conference on “Laws of Chaos” (http://ricardo.ecn.wfu.edu/~cottrell/OPE/archive/0705/att-0184/01-CFP.txt) and was very happy to meet both Emmanuel Farjoun and Mosche Machover. The conference was a great success, but unfortunately the conference web-page and accompanying papers is no longer maintained. My own work on probabilistic macroeconomic models (e.g., https://ianwrightsite.files.wordpress.com/2017/10/wright-social-architecture-capitalism.pdf and http://www.economics-ejournal.org/economics/journalarticles/2009-19) is very much in the tradition of Marxist probabilistic political economy (e.g., see http://eprints.lse.ac.uk/36428/1/The_stochastic_concept_of_economic_equilibrium_a_radical_alternative_(lsero).pdf). To make contact with empirical data then the probabilistic approach is necessary. Also, the concept of a statistical equilibrium is clearly more suited to describing economic systems compared to deterministic equilibrium (which is the typical case in both maintstream and Marxist models). Lots more could be said here.

      Like

  3. I was going over the math where you solve the simultaneous equations to derive x=-cos(t) and y=sin(t), when I noticed you didn’t mention the boundary conditions (from dividing by sin/cos) of sin(t) =/= 0 and cos(t) =/= 0. I guess these simultaneously represent the inseparability of being and nothing as one can’t exist without the other, as well as that pure being and pure nothing are both impossible and identical.

    Additionally, I noticed you missed the minus before k1^2 in (k3+1)^2 = k1^2 and by extension the i before k1 in k1=k3+1, but it doesn’t change the unique solutions for x (being) and y (nothing) where both are simultaneously real (when k1=0). However, it does mean there are unique solutions where x is real (whenever k1 is non-zero and real) with y being complex (partially imaginary), and where y is real (whenever k1 is non-zero and complex) with x being complex (partially imaginary). To understand whether these extra solutions are relevant, you would have to either; determine what k1 physically means (for possible boundary conditions), or prove that k1=0, or prove that x and y cant be complex.

    Alternatively, you could use different initial conditions, where x(0)=0 represents “being” at “pure nothing” then “coming-to-be”, and where y(0)=1 represents “nothing” at “pure being” then “ceasing-to-be”. This gives solutions of x=sin(t) and y=cos(t) without any boundary conditions on sin(t) or cos(t).

    Like

    1. Thanks KM. On the specific point of initial conditions I set x(0)=1 and y(0)=0 as initial conditions; see the notes captioned, “The solution to Hegel’s equations, which describe how being and nothing change when sublated as a process of becoming.” I will think further about your other points! It would be nice to derive the equations of motion from even deeper principles and thereby avoid some of the arbitrariness of specifying initial conditions. Thanks for engaging! Best wishes, Ian.

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